21  Session 12: The Exit Boundary \(L^*(t)\) — Numerical Examples

Unit 3 — Decision and Application
Book Chapter 6 (sections 6.5–6.7)
Track Common core (both tracks)

21.1 Learning Objectives

By the end of this session, students will be able to:

  1. Read an exit boundary \(L^*(t)\) chart and identify the hold region vs. exit region.
  2. Explain why \(L^*(t)\) rises as \(t \to T\) (the boundary terminal-time effect).
  3. Compute the optimal exit decision for a given (current \(L\), remaining \(T\)) pair using the calibrated boundary.
  4. Identify how the exit boundary differs across asset classes (PE buyout vs. infrastructure vs. private credit).
  5. Apply the exit boundary to one of the major historical episodes (GFC 2009, COVID 2020) and identify what GE-LAV would have recommended.

21.2 Pre-Class Assignment

  • Read: Book Chapter 6, sections 6.5–6.7 (~8 pages)
  • Try: Sketch on paper what you think the exit boundary looks like — does \(L^*\) rise or fall as \(t\) increases? Why?

21.3 In-Class Outline (75 minutes)

Time Segment Format
0:00–0:10 Recap Session 11 · Set up today’s numerical work Lecture
0:10–0:25 The exit boundary chart — reading the figure Lecture
0:25–0:40 Why \(L^*(t)\) has the shape it does Lecture + chart deep-dive
0:40–0:55 Sensitivity to OU parameters and asset class Lecture
0:55–1:10 Historical application: GFC, COVID, rate shock Case + discussion
1:10–1:15 Worked example: should this LP exit? Live problem

21.4 Discussion Questions

  1. CalPERS publicly disclosed 2009 PE secondary sales at 25% discount. Using the exit boundary chart, would GE-LAV have recommended this sale? What information beyond the chart would CalPERS have needed?
  2. The exit boundary \(L^* \approx -0.5\) at 1-year horizon. Most fund extensions are 1-2 years. Does the boundary suggest fund extensions should be granted (hold) or refused (force exit)? Why?
  3. If your investment policy statement (IPS) says “no secondary sales except in distress,” is that consistent with GE-LAV-optimal behavior? When would it diverge?

21.5 Worked Numerical Example: Should This LP Exit?

Scenario: It is Q4 2024. An insurance company holds a 9-year-old infrastructure fund position. Reported NAV = $200M. Position has 4 years remaining until natural liquidation.

Current conditions: - Estimated \(L_t = -0.7\) (moderately stressed; secondary discounts around 18%) - Asset class: infrastructure - Remaining horizon: 4 years - Secondary market bid: 22% discount → $156M cash today - Solvency II capital charge if held: increases by ~3% of NAV - Solvency II capital charge if sold: decreases by ~5% of NAV (cash is treated favorably)

Step 1: What does the exit boundary say?

For infrastructure at 4 years remaining: \(L^* \approx -1.5\)

Current \(L_t = -0.7\) vs. \(L^* = -1.5\)

\(L_t > L^*\), so the GE-LAV recommendation is HOLD.

Step 2: But Solvency II is forcing pressure

The capital charge differential (3% + 5% = 8% effective swing) creates a regulatory wedge that GE-LAV doesn’t directly model. If the insurer would be capital-constrained by holding: - Hold cost: 8% of $200M = $16M of capital tied up - Sell cost: 22% discount = $44M of value sacrificed

Net: Selling sacrifices $44M − $16M = $28M of value vs. holding through.

Step 3: GE-LAV expected value of holding

Under OU dynamics, \(E[L_{t+4}] = \bar{L} + (L_t - \bar{L}) e^{-4\kappa} = 1.0 + (-0.7 - 1.0) e^{-1.8} = 1.0 + (-1.7)(0.165) = 0.72\)

The expected liquidity state in 4 years is +0.72, well above current. The Jensen bias for 4-year infrastructure: \(B(4) = 0.18\% \times 4 = 0.72\%\)

Expected hold value: \(\$200M \times (1 + 0.0072) \approx \$201.4M\), plus mean-reversion gain (state goes from −0.7 to +0.7 generates ~5–8% NAV uplift via lower discount rates).

Conclusion: Pure GE-LAV says hold. Hybrid view (GE-LAV + Solvency II): probably still hold, but the case is closer than a non-regulatory LP would face.

21.6 What to Expect Next Session

Session 13 covers liquidity traps and the historical performance of the GE-LAV exit rule across multiple episodes. We’ll examine:

  • The 2008–2010 GFC in detail (the worst-case test of the framework)
  • COVID 2020 (sharp shock, fast recovery — does GE-LAV handle this?)
  • Rate shock 2022 (slow-burn vs. fast crisis)
  • The Pigouvian exit tax (preview — full treatment Session 24)

Reading: Book Chapter 6, sections 6.7–6.10 (~10 pages).

← Session 11 | Schedule | Next: Session 13 →